| Version 5 |
Version 4 |
| Let X be a topological space, and let $Y$ be the adjunction $Y := X\cup_\varphi D^k,$ where $D^k$ is a closed $k$-ball and $\funcdef{\varphi}{S^{k-1}}{X}$ is a continuous map, with $S^{k-1}$ is the $(k-1)$-sphere considered as the boundary of $D^k.$ Then, we say that $Y$ is obtained from $X$ by the {\em attachment of a $k$-cell.} |
Let X be a topological space, and let $Y$ be the adjunction $Y := X\cup_\varphi D^k,$ where $D^k$ is a closed $k$-ball and $\funcdef{\varphi}{S^{k-1}}{X}$ is a continuous map, with $S^{k-1}$ is the $(k-1)$-sphere considered as the boundary of $D^k.$ Then, we say that $Y$ is obtained from $X$ by the {\em attachment of a $k$-cell.} |
| The image $e^k$ of $D^k$ in $Y$ is called a {\em closed $k$-cell,} and the image $\oce^k$ of the interior $\ocD := D^k\setminus S^{k-1}$ of $D^k$ is the corresponding {\em open $k$-cell.} |
The image $e^k$ of $D^k$ in $Y$ is called a {\em closed $k$-cell,} and the image $\oce^k$ of the interior $\ocD := D^k\setminus S^{k-1}$ of $D^k$ is the corresponding {\em open $k$-cell.} |
| Note that for $k=0$ the above definition reduces to the statement that $Y$ is the disjoint union of $X$ with a one-point space. |
Note that for $k=0$ the above definition reduces to the statement that $Y$ is the disjoint union of $X$ with a one-point space. |
| More generally, we say that $Y$ is obtained from $X$ by {\em cell attachment\/} |
More generally, we say that $Y$ is obtained from $X$ by {\em cell attachment\/} |
| if $Y$ is homeomorphic to an adjunction $X\cup_\set{\varphi_i} D^{k_i},$ where the maps $\set{\varphi_i}$ into $X$ are defined on the boundary spheres of closed balls $\set{D^{k_i}}.$ |
if $Y$ is homeomorphic to an adjunction $X\cup_\set{\varphi_i} D^{k_i},$ where the maps $\set{\varphi_i}$ into $X$ are defined on the boundary spheres of closed balls $\set{D^{k_i}}.$ |
| %\begin{rmk} |
\begin{rmk} |
| %A recognition principle for attached cells is as follows: Let $Y$ be a |
A recognition principle for attached cells is as follows: Let $Y$ be a Hausdorff topological space and $e$ a closed subspace such that there exists a map $\funcdef{\Phi}{D^k}{Y}, k\ge 1,$ satisfying: |
| %Hausdorff topological space and $e$ a closed subspace such that there exists a |
\begin{enumerate} |
| %map $\funcdef{\Phi}{D^k}{Y}, k\ge 1,$ satisfying: |
\item |
| %\begin{enumerate} |
$\Phi(D^k) = e$ and |
| %\item |
\item |
| %$\Phi(D^k) = e$ and |
the restriction of $\Phi$ to $\ocD^k := D^k\setminus \bdry D^k$ is an embedding. |
| %\item |
\end{enumerate} |
| %the restriction of $\Phi$ to $\ocD^k := D^k\setminus \bdry D^k$ is an |
Then, $Y$ is obtained from $X := Y\setminus\Phi(\ocD^k)$ by the attachment of the $k$-cell $e$. $k$ is called the {\em dimension} of $e,$ and is well-defined by virtue of the invariance of domain theorem. |
| %embedding. |
Attached $0$-cells are recognized as being isolated points of $X$. |
| %\end{enumerate} |
\end{rmk} |
| %Then, $Y$ is obtained from $X := Y\setminus\Phi(\ocD^k)$ by the attachment of 5 |
|
| %the $k$-cell $e$. $k$ is called the {\em dimension} of $e,$ and is well-defined %by virtue of the invariance of domain theorem. |
|
| %Attached $0$-cells are recognized as being isolated points of $X$. |
|
| %\end{rmk} |
|
